Solve 3/5sqrt{500}div3/2ksqrt{20}

$$\frac { 3 } { 5 } \sqrt { 500 } \div \frac { 3 } { 2 } k \sqrt { 20 }$$

Evaluate

$$40k$$

Solution Steps

Factor 500=10^{2}\times 5. Rewrite the square root of the product \sqrt{10^{2}\times 5} as the product of square roots \sqrt{10^{2}}\sqrt{5}. Take the square root of 10^{2}.

$$\frac{\frac{3}{5}\times 10\sqrt{5}}{\frac{3}{2}}k\sqrt{20}$$

Express \frac{3}{5}\times 10 as a single fraction.

$$\frac{\frac{3\times 10}{5}\sqrt{5}}{\frac{3}{2}}k\sqrt{20}$$

Multiply 3 and 10 to get 30.

$$\frac{\frac{30}{5}\sqrt{5}}{\frac{3}{2}}k\sqrt{20}$$

Divide 30 by 5 to get 6.

$$\frac{6\sqrt{5}}{\frac{3}{2}}k\sqrt{20}$$

Divide 6\sqrt{5} by \frac{3}{2} by multiplying 6\sqrt{5} by the reciprocal of \frac{3}{2}.

$$\frac{6\sqrt{5}\times 2}{3}k\sqrt{20}$$

Multiply 6 and 2 to get 12.

$$\frac{12\sqrt{5}}{3}k\sqrt{20}$$

Divide 12\sqrt{5} by 3 to get 4\sqrt{5}.

$$4\sqrt{5}k\sqrt{20}$$

Factor 20=2^{2}\times 5. Rewrite the square root of the product \sqrt{2^{2}\times 5} as the product of square roots \sqrt{2^{2}}\sqrt{5}. Take the square root of 2^{2}.

$$4\sqrt{5}k\times 2\sqrt{5}$$

Multiply 4 and 2 to get 8.

$$8\sqrt{5}k\sqrt{5}$$

Multiply \sqrt{5} and \sqrt{5} to get 5.

$$8\times 5k$$

Multiply 8 and 5 to get 40.

$$40k$$

Differentiate w.r.t. k

$$40$$

Steps Using Definition of a Derivative

Factor 500=10^{2}\times 5. Rewrite the square root of the product \sqrt{10^{2}\times 5} as the product of square roots \sqrt{10^{2}}\sqrt{5}. Take the square root of 10^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}k}(\frac{\frac{3}{5}\times 10\sqrt{5}}{\frac{3}{2}}k\sqrt{20})$$

Express \frac{3}{5}\times 10 as a single fraction.

$$\frac{\mathrm{d}}{\mathrm{d}k}(\frac{\frac{3\times 10}{5}\sqrt{5}}{\frac{3}{2}}k\sqrt{20})$$

Multiply 3 and 10 to get 30.

$$\frac{\mathrm{d}}{\mathrm{d}k}(\frac{\frac{30}{5}\sqrt{5}}{\frac{3}{2}}k\sqrt{20})$$

Divide 30 by 5 to get 6.

$$\frac{\mathrm{d}}{\mathrm{d}k}(\frac{6\sqrt{5}}{\frac{3}{2}}k\sqrt{20})$$

Divide 6\sqrt{5} by \frac{3}{2} by multiplying 6\sqrt{5} by the reciprocal of \frac{3}{2}.

$$\frac{\mathrm{d}}{\mathrm{d}k}(\frac{6\sqrt{5}\times 2}{3}k\sqrt{20})$$

Multiply 6 and 2 to get 12.

$$\frac{\mathrm{d}}{\mathrm{d}k}(\frac{12\sqrt{5}}{3}k\sqrt{20})$$

Divide 12\sqrt{5} by 3 to get 4\sqrt{5}.

$$\frac{\mathrm{d}}{\mathrm{d}k}(4\sqrt{5}k\sqrt{20})$$

Factor 20=2^{2}\times 5. Rewrite the square root of the product \sqrt{2^{2}\times 5} as the product of square roots \sqrt{2^{2}}\sqrt{5}. Take the square root of 2^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}k}(4\sqrt{5}k\times 2\sqrt{5})$$

Multiply 4 and 2 to get 8.

$$\frac{\mathrm{d}}{\mathrm{d}k}(8\sqrt{5}k\sqrt{5})$$

Multiply \sqrt{5} and \sqrt{5} to get 5.

$$\frac{\mathrm{d}}{\mathrm{d}k}(8\times 5k)$$

Multiply 8 and 5 to get 40.

$$\frac{\mathrm{d}}{\mathrm{d}k}(40k)$$

The derivative of ax^{n} is nax^{n-1}.

$$40k^{1-1}$$

Subtract 1 from 1.

$$40k^{0}$$

For any term t except 0, t^{0}=1.

$$40\times 1$$

For any term t, t\times 1=t and 1t=t.

$$40$$